C₂²×#576 #63084

order	256
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂²×#576

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁶
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

5731 conjugacy classes of subgroups; only the 3143 normal subgroups are shown.

normal subgroups only

This group has 5731 conjugacy classes of subgroups — too many to draw. Only its 3143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2);(3,4);(5,6)(7,11)(8,10)(9,13)(12,16)(14,18)(15,17)(19,20);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19);(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20);(7,14)(11,18)(15,20)(17,19)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#576, C₂²×#576⋊C₃, #11710886
maximal subgroup of	#359609, #359613, #359674, …
maximal normal subgroup of	#359609, #359613, #359674, …
maximal quotient of	C₂²×#2897, C₂²×#2913, C₂²×#2946, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56072);